Note that Definition REF  does not require the dynamics to be sparse. In fact, matrices \(A\)  and \(B\)  of the form in Definition REF  can be arbitrarily dense, i.e., the evolution of a subsystem state \(x_i(t+1)\)  can depend on subsystems that \(i\)  cannot directly communicate with [1]}. This is distinct from sparsity/delay constraints used in [2]}, [3]}, [4]}, and encodes a different notion of symmetry than that exploited in the distributed control of spatially invariant systems [5]}.
